Question:

<div class="solution"><p><strong>Step 1:</strong> Domain of <span class="math-inline">$g\circ f$</span>: need <span class="math-inline">$e^x\in[-1,1]$</span>. Since <span class="math-inline">$e^x>0$</span>, need <span class="math-inline">$e^x\le 1\implies x\le 0$</span>. Domain = <span class="math-inline">$(-\infty,0]$</span> ✓ (C true, A false).</p><p><strong>Step 2:</strong> Range of <span class="math-inline">$g\circ f$</span>: <span class="math-inline">$e^x\in(0,1]$</span> → <span class="math-inline">$\sin^{-1}(e^x)\in(0,\pi/2]$</span>. This is a subset of range of g = <span class="math-inline">$[-\pi/2,\pi/2]$</span> ✓ (B true).</p><p><strong>Step 3:</strong> Range is <span class="math-inline">$(0,\pi/2]$</span>, not <span class="math-inline">$[-\pi/2,0]$</span> (D false).</p><p><strong>Answer: (B),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> Forgetting eˣ>0 always, so the sin⁻¹ output is always positive.</div><div class="key-concept"><strong>Key Concept:</strong> Range of inner exponential restricts which branch of outer sin⁻¹ is accessed</div></div>